One inexpensive herbicide is a combination of vinegar and dish soap. Apply this treatment on a sunny day. The acetic acid in vinegar will burn the weed leaves on contact and lower the pH of the soil, making recovery difficult.
To a gallon of vinegar, add a teaspoon of liquid dish soap,
which helps the vinegar stick to the plant leaves. If you plan to use the area treated for something other than plants, you can add 1/2 cup of salt as well. This will add a final blow to the ground. Remember in history and Bible classes, reading about how a conquerer salted the fields? Just in case there were battle survivors, plants would not grow for some time and the survivors would starve.